Are you familiar with the tight binding model? when matter forms the condensed state, it forms bonds. A solid can be thought as a macromolecule. The valence electrons no longer "belong" to specific atoms, they are delocalized in collective orbitals. Those orbitals give rise to the band structure. In particular, bands associated to the sp hybrid orbitals are 'sp' bands. You should look for the tight binding model in a solid state physics textbook.